We study the $O(N)$ non-linear $\sigma$ model on three-dimensional manifolds of constant curvature by means of the large $N$ expansion at the critical point. We examine saddle point equations imposing anti-periodic boundary condition in time direction. In the case $S^1 \times S^2$ we find that a solution is inevitably unstable. We briefly refer to the case $S^1 \times S^1 \times S^1$.
A Weak Power-Counting Theorem for the Renormalization of the Non-Linear Sigma Model in Four Dimensions
Non-Canonical Perturbation Theory of Non-Linear Sigma Models
Operator Product Expansions in the Two-Dimensional O(N) Non-Linear Sigma Model
